Abstract. Lemma speculation has long been considered a promising technique to automate the discovery of missing lemmas for inductive proofs. This technique involves speculating a schematic lemma that becomes incrementally instantiated by unification as the proof continues. This synthesis process is known as middle-out reasoning. We have extended lemma speculation, and more generally middle-out reasoning, to dynamic rippling for higher-order domains, implemented it in the Isa-Planner system and improved the technique to ensure termination. This provides a practical basis for exploring the applications of middle-out reasoning. We demonstrate such an application by performing a critical and comparative evaluation of lemma speculation. This shows that when lemma speculation is applied it often finds the needed lemmas to complete the proof, but it is not applicable as often as initially expected. In comparison, we show that simpler proof methods combined with theory formation methods offer an effective alternative. 1

Abstract. We present a succinct account of dynamic rippling, a technique used to guide the automation of inductive proofs. This simplifies termination proofs for rippling and hence facilitates extending the technique in ways that preserve termination. We illustrate this by extending rippling with a terminating version of middle-out reasoning for lemma speculation. This supports automatic speculation of schematic lemmas which are incrementally instantiated by unification as the rippling proof progresses. Middle-out reasoning and lemma speculation have been implemented in higher-order logic and evaluated on typical libraries of formalised mathematics. This reveals that, when applied, the technique often finds the needed lemmas to complete the proof, but it is not as frequently applicable as initially expected. In comparison, we show that theory formation methods, combined with simpler proof methods, offer an effective alternative. 1